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Complete asymptotic expansions for eigenvaluesofDirichlet Laplacian in thin three-dimensional rods*

Published online by Cambridge University Press:  06 August 2010

Denis Borisov
Affiliation:
Bashkir State Pedagogical University, October Revolution St. 3a, 450000 Ufa, Russia. [email protected]
Giuseppe Cardone
Affiliation:
University of Sannio, Department of Engineering, Corso Garibaldi, 107, 82100 Benevento, Italy. [email protected]
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Abstract

We consider the Dirichlet Laplacian in a thin curvedthree-dimensional rod. The rod is finite. Its cross-section isconstant and small, and rotates along the reference curve in anarbitrary way. We find a two-parametric set of the eigenvalues ofsuch operator and construct their complete asymptotic expansions. Weshow that this two-parametric set contains any prescribed number ofthe first eigenvalues of the considered operator. We obtain thecomplete asymptotic expansions for the eigenfunctions associatedwith these first eigenvalues.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2010

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References

N.S. Bakhvalov and G.P. Panasenko, Homogenization: Averaging processes in periodic media. Kluwer, Dordrecht/Boston/ London (1989).
D. Borisov and P. Freitas, Singular asymptotic expansions for Dirichlet eigenvalues and eigenfunctions on thin planar domains. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26 (2009) 547–560. CrossRef
Bouchitté, G., Mascarenhas, M.L. and Trabucho, L., On the curvature and torsion effects in one dimensional waveguides. ESAIM: COCV 13 (2007) 793808. CrossRef
G. Cardone, T. Durante and S.A. Nazarov, The localization effect for eigenfunctions of the mixed boundary value problem in a thin cylinder with distorted ends. SIAM J. Math. Anal. (to appear).
Duclos, P. and Exner, P., Curvature-induced bound states in quantum waveguides in two and three dimensions. Rev. Math. Phys. 7 (1995) 73102. CrossRef
Freitas, P. and Krejčiřík, D., Location of the nodal set for thin curved tubes. Indiana Univ. Math. J. 57 (2008) 343376. CrossRef
L. Friedlander and M. Solomyak, On the spectrum of the Dirichlet Laplacian in a narrow infinite strip, in Spectral theory of differential operators: M. Sh. Birman 80th anniversary collection, Adv. Math. Sci. 225, T. Suslina and D. Yafaev Eds., AMS Translations – Series 2, Providence (2008).
Friedlander, L. and Solomyak, M., On the spectrum of the Dirichlet Laplacian in a narrow strip. Israel J. Math. 170 (2009) 337354. CrossRef
D. Grieser, Thin tubes in mathematical physics, global analysis and spectral geometry, in Analysis on Graphs and Its Applications, P. Exner, J.P. Keating, P. Kuchment, T. Sunada and A. Teplyaev Eds., Proc. Symp. Pure Math. 77, AMS, Providence (2008).
Krejčiřík, D., Spectrum of the Laplacian in a narrow curved strip with combined Dirichlet and Neumann boundary conditions. ESAIM: COCV 15 (2009) 555568. CrossRef
V.P. Mikhajlov, Partial differential equations. Moscow, Mir Publishers (1978).
S.A. Nazarov, Dimension Reduction and Integral Estimates, Asymptotic Theory of Thin Plates and Rods 1. Novosibirsk, Nauchnaya Kniga (2001).
O.A. Oleinik, A.S. Shamaev and G.A. Yosifyan, Mathematical problems in elasticity and homogenization, Studies in Mathematics and its Applications 26. Amsterdam etc., North-Holland (1992).
Panasenko, G.P. and Perez, M.E., Asymptotic partial decomposition of domain for spectral problems in rod structures. J. Math. Pures Appl. 87 (2007) 136. CrossRef
Vishik, M.I. and Lyusternik, L.A., The asymptotic behaviour of solutions of linear differential equations with large or quickly changing coefficients and boundary conditions. Russ. Math. Surv. 15 (1960) 2391. CrossRef